Lesson 3 - 3

1. Lesson 3 - 3 Correlation and Regression Wisdom

2. Knowledge Objectives • Recall the three limitations on the use of correlation and regression. • Explain what is meant by an outlier in bivariate data. • Explain what is meant by an influential observation and how it relates to regression. • Define a lurking variable. • Give an example of what it means to say “association does not imply causation.”

3. Construction Objectives • Given a scatterplot in a regression setting, identify outliers and influential observations • Explain how correlations based on averages differ from correlations based on individuals

4. Vocabulary • Influential Observation – an observation that if removed would markedly change the result of the regression calculation

5. Limitations • Correlation and regression describe only linear relationships • Extrapolation (using model outside range of the data) often produces unreliable predications • Correlation is not resistant (to outliers!)

6. Outliers vs Influential Observation • Outlier is an observation that lies outside the overall pattern of the other observations • Outliers in the Y direction will have large residuals. but may not influence the slope of the regression line • Outliers in the X direction are often influential observations • Influential observation is one that if by removing it, it would markedly change the result of the regression calculation

7. Example 1 Does the age at which a child begins to talk predict later score on a test of metal ability? A study of the development of 21 children recorded the age in months at which they spoke their first word and their later Gesell Adaptive Score (GAS).

8. Example 1 cont • What is the equation of the LS regression line used to model this data? • What is the interpretation of this data? y-hat = 109.8738 – 1.127x r = -0.64 The scatter plot and the slope of the regression line indicates a negative association. Children who begin to speak later tend to have lower test scores than early talkers. The slope suggests that for every month older a child is when they begin to speak, their score on the Gesell test will decrease by about 1.13 points. The y-intercept has no real meaning in this case.

9. Example 1 cont • Are there any outliers? • Are there any influential observations? Child #19 is an outlier in the Y-direction and child #18 is an outlier in the X-direction. Child #18 is an outlier in the X-direction and also an influential observation because it has a strong influence on the positioning of the regression line.

10. Example 1 cont Scatterplot w/ Regression Line Residual Plot

11. Lurking or Extraneous Variable • The relationship between two variables can often be misunderstood unless you take other variables into account • Association does not imply causation! • Instances of Rocky Mt spotted fever and drownings reported per month are highly correlated, but completely without causation

12. Remember Sampling Distributions • When we looked at individual values, they had much broader spreads (variances) than when we looked at the distributions of x-bar • Same is true with correlations based on averaged data – strong correlations may exist between averages, but individuals will have much greater variances • Correlations based on averages are usually too high when applied to individuals.

13. Summary and Homework • Summary • Correlation and regression must be interpreted with caution • Plot data to be sure that the relationship is roughly linear and to detect outliers • Check for influential observations that substantially change the regression line • Lurking variables may explain the relationship between the explanatory and response variables • Homework • pg 242-3 3.63-67